#6820919

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1215 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,7)(3,6,4,8)(9,10)(11,18)(12,21)(13,16)(14,17)(15,20)(19,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,18)(21,24)(26,30)(27,33)(28,34)(31,36)(32,38)(35,39);(1,4,5,3)(2,6,7,8)(25,26,28,30,29,31,34,36)(27,32,33,37,35,38,39,40);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,31)(30,36)(32,38)(37,40);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,27,34,39,29,35,28,33)(26,32,36,40,31,38,30,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6728115);